Unit-Regularity and Stable Range Conditions#

Abstract

ABSTRACT A ring R has stable range one (respectively, weakly stable range one) provided whenever ax  +  b  = 1 in R , there exists y  ∈  R such that a  +  by is a two-sided (respectively, one-sided) inverse. In the first part of this article, we show that an exchange ring R has stable range one if and only if for every regular element x  ∈  R there exist some unit-regular element w  ∈  R and some idempotent e  ∈  R such that x  =  ew if and only if whenever aR  +  bR  =  dR in R , there exists a unit-regular element w  ∈  R such that a  +  bz  =  dw for some z  ∈  R . These generalize a result of Canfell (1995, Theorem 2.9) and a special case of Chen (2000b, Theorem 4.2). In the second part, we prove that a module M satisfies the internal weak cancellation property if and only if every regular element of the endomorphism ring of M is one-sided unit-regular. We also extend a well known result on exchange rings with stable range one to exchange rings with weakly stable range one, by showing that an exchange ring R has weakly stable range one if and only if whenever in R , there exists a one-sided inverse u  ∈  R such that au  =  ub .